The summary of ‘LLM Control Theory Seminar (April 2024)’

The video discusses the development and application of a control theory for language models, focusing on understanding, manipulating, and optimizing their output. Key points include the creation of a control theoretic framework, experiments on model controlability, the importance of control inputs and prompts, and exploring zero-shot learning and generalization in language models. The video delves into self-attention mechanisms, normalization layers in Transformers, and the concept of control theory in deep learning systems. It also covers K Epsilon controllability, the concept of reachability, and the role of control inputs in achieving desired outputs. The speaker explores the complexity of language models, the concept of zero temperature sampling, and the importance of bounded orthogonality in controlling the model's output. Future directions involve merging theory with experimental results, prompt optimization methods, and exploring controllable subspaces in language model activations.

00:00:00

In this part of the video, Manal discusses the motivation behind building a control theory of language models, explaining the benefits of understanding and applying language models better. She outlines the control theoretic framework they created, discussing reachability and control in self-attention heads. Manal also mentions a theorem they proved on self-attention controllability, along with experiments they conducted on model controlability. Additionally, she highlights open questions in language model control theory, emphasizing the tractable nature of these questions. Manal also touches on the zero-shot learning ability of language models, their generalization capabilities, and the impact of prompts on model performance. She gives a brief overview of how language models predict the next token and the working mechanism of Transformer-based language models using self-attention and token representations.

00:05:00

In this segment of the video, the speaker discusses how normalization layers are used in Transformers to generate probability distributions over the next token. The forward flow of information in Transformers is highlighted, leading to high throughput in autoregressive language models. The video emphasizes the control theory approach in understanding these language models, discussing the need for control inputs and prompt optimization methods to guide the model’s output. The example given involves manipulating prompts to influence the generated text output. The analogy of cruise control in control theory is used to explain the concept of control inputs guiding system behavior. This raises questions about the number of control tokens needed to direct the language model output effectively, sparking interest in developing a control theory for language models.

00:10:00

In this segment of the video, the speaker discusses the mathematical theory and practical applications of control systems. They explain how controlling systems involves manipulating poles in linear time-invariant systems to change system behavior. The discussion goes into controllability tests and system decompositions. The speaker also mentions how language models, like OpenAI’s chatbot, can be viewed as control systems. They give examples of using control inputs in language models to achieve desired outputs while preventing unwanted behavior. The idea of control inputs is applied to developing systems like fang.com for predicting job success based on resume information. The segment also explores the increasing complexity of language model systems with feedback loops and agents.

00:15:00

In this segment of the video, the speaker discusses the complexity of language models by proposing a systems or control theory understanding approach. They introduce the concept of formalizing language models as systems, examining controllability, and developing theory and engineering practices to build safe and effective language model systems. The speaker acknowledges the uniqueness of language model systems compared to conventional dynamical systems due to their discrete state and time characteristics, sequential token operation, state dynamics related to token input/output, and mutual exclusion of input and generation. The formalization of language model systems involves vocabulary sets, probability distributions for token sequences, and control inputs guiding state updates and token generation. The goal is to leverage control theory insights to enhance the development of language models effectively and safely.

00:20:00

In this part of the video, the speaker discusses the concept of zero temperature sampling in language models. They explain the deterministic nature of the system and how it can be analyzed experimentally and theoretically. The discussion includes the update function for sampling X Prime, which involves taking the argmax of the maximum likelihood next token at each time point. Additionally, the speaker introduces a control-theoretic perspective on language models, emphasizing the relationship between the initial state, control input, and system output. They delve into the concepts of reachability and control, highlighting how reachability is defined in terms of output token sequences being reachable from the initial state within a certain time frame. The speaker then touches on the notion of reachable sets, explaining that it refers to the set of all outputs reachable from a given initial condition using a specified number of control tokens. The importance of considering the minimal length control input for achieving objectives is also emphasized.

00:25:00

In this part of the video, the speaker discusses K Epsilon controllability in the context of system control. Usually, controllability implies that the reachable set from any initial condition covers the entire output space. However, K Epsilon controllability introduces constraints to measure the fraction of reachable outputs using a limited control sequence length (K tokens). The goal is to minimize the probability (Epsilon) that any output is unreachable as the number of control tokens increases. The discussion also touches on self-attention controllability in deep learning systems, detailing how information is transmitted between token representations using self-attention matrices in Transformer models.

00:30:00

In this segment of the video, the speaker explains the concept of self-attention in neural networks. They discuss the computation involved in self-attention using query and key projection matrices, soft max normalization, and introduce an alternative notation for softmax as a matrix multiplication. The speaker then delves into the idea of control in self-attention, where the goal is to select control inputs that lead to the desired output. They talk about partitioning inputs and outputs into controllable and uncontrollable components and define the notion of control. Additionally, the speaker mentions the generalization of these notions to tasks like token generation. They touch on the concept of normalized attention matrices and splitting the output into controllable and uncontrollable parts. The speaker also introduces a theorem related to control inputs and desired outputs, highlighting a necessary condition for reaching the desired output. Further explanations and proofs related to control in self-attention are presented, focusing on canceling out orthogonal components to reach the desired output.

00:35:00

In this segment of the video, the speaker explains the concept of YX Min as the rescaled component of Y corresponding to X with maximal attention given to the control input U. The role of the attention parameter matrices and the significance of beta in controlling the orthogonal component of the Y matrix are discussed. The video also touches on how YX lies on a line segment between a minimal and a maximal part, with the importance of bounded orthogonality. The speaker further elaborates on the token space in the context of Transformer models, discussing the embedding of tokens, the generation of next tokens, and the transition from discrete to continuous vector representations within the model layers.

00:40:00

In this segment of the video, the speaker discusses the next steps in situating a theorem within LLMS by merging it with experimental results. The focus is on controlling output representations to control generation from models. They introduce the concept of K Epsilon controllability and outline how to measure it practically using data sets and prompt optimization methods. The experiments involve sampling token sequences from Wikipedia and measuring controllability with different data sets, including real language, top likely tokens, and random tokens. Two algorithms, exhaustive greedy search and greedy coordinate gradient, are used to find optimal prompts. The goal is to determine the reachability of output tokens with a limited number of control inputs.

00:45:00

In this part of the video, the speaker discusses utilizing methods like stochastic local search and greedy coordinate gradient to optimize prompts iteratively. They present results showing the ability to achieve the desired output over 97% of the time by using at most 10 tokens. The discussion touches on controlling models to predict the correct next token and the importance of prompt length in achieving better control. They mention challenges with longer initial state sequences and introduce the concept of distributional control for better understanding the probability distribution over next tokens. The speaker expresses interest in exploring typical sequences for trajectory selection to enhance model controllability.

00:50:00

In this part of the video, the speaker discusses control theory and information theory, emphasizing the importance of utilizing typical sequence sets with high probabilities in language modeling. The concept of Chain of Thought is introduced, highlighting its role in generating intermediate tokens to improve question answering by explaining reasoning. The speaker also suggests investigating sensitivity analysis, learnability, and computational costs in controlling language models. They mention the idea of composing networks of language models for information exchange as a way to advance the field. Additionally, the discussion touches upon exploring controllable subspaces in language model activations, including emotional representations. The speaker expresses gratitude to mentors and collaborators for their contributions to the research. Finally, a question is raised regarding the challenge of ensuring efficient control in language models despite potential distractions from imposed tokens.